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Exact dimerized phase in anisotropic XYZ model for quasi-one-dimensional magnets

Hong-Ze Xu, Shunyao Zhang, Guang‐Can Guo, Ming Gong

Published Jun 15, 2018• 0 citations
OpenAlex

Abstract

We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by direct generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic regime. This model also supports three different ferromagnetic (FM) phases, denoted as $x$-FM, $y$-FM and $z$-FM. The boundaries between the dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry or $\mathbb{Z}_2$ symmetry lead to dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge $c=1$. We characterize the properties of these boundaries using entanglement entropy, excitation gap and long-range spin-spin correlations. These results are relevant to a large number of quasi-one-dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in true materials with magnetic field along different directions, and show that the anisotropy may resolve the long-standing disagreements between theory and experiments.

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Exact dimer phase with anisotropic interaction for one dimensional magnets

Hong-Ze Xu, Shun-Yao Zhang, Guang-Can Guo +1 more

We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic region. We demonstrate that this phase can be realized even in models when antiferromagnetic interaction along one of the three directions. This model also supports three different ferromagnetic (FM) phases, denoted as x-FM, y-FM and z-FM, polarized along the three directions. The boundaries between the exact dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry breaking or [Formula: see text] symmetry breaking leads to exact dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge [Formula: see text] for free fermions. We characterize the properties of these boundaries using entanglement entropy, excitation gap, and long-range spin-spin correlation functions. These results are relevant to a large number of one dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in realistic materials with magnetic field along different directions and show that the anisotropy may resolve the disagreement between theory and experiments based on isotropic spin-spin interactions.

arXiv:1806.05814•Scientific Reports•Mar 19, 2021•11 citations

Exact dimer ground states of long-range spin chains and ladders

Jędrzej Wardyn, Miłosz Panfil

Interacting spin chains and ladders are known to support a plethora of quantum phases with complex ground-state phase diagrams. In this work, we study a large family of such models and determine precise, explicit conditions under which an exact dimer state is guaranteed to be the ground state. These general conditions are validated for various generalizations of the Majumdar-Ghosh model using exact diagonalization. Our results provide exact reference points in the phase diagrams of a wide class of spin chains and ladders, including those with anisotropic and arbitrary-range interactions.

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Quantum phase transitions of one-dimensional period-two anisotropic XY models in a transverse field

Song Jia-Li, Ming Zhong, Tong Pei-Qing

The quantum phase transitions of one-dimensional period-two anisotropic XY models in a transverse field with the Hamiltonian where the anisotropy parameters i take and alternately, are studied. The Hamiltonian can be reduced to the diagonal form by Jordan-Wigner and Bogoliubov transformations. The long-range correlations Cx and Cy are calculated numerically. The phase with Cx Cy0 (or Cy Cx0) is referred to as the ferromagnetic (FM) phase along the x (or y) direction, while the phase with Cx=Cy=0 is the paramagnetic (PM) phase. It is found that the phase diagrams with the ratio -1 and =-1 are different obviously. For the case with -1, the line h=hc1=1-[(1-)/2]2 separates an FM phase from a PM phase, while the line =0 is the boundary between a ferromagnetic phase along the x direction (FMx) and a ferromagnetic phase along the y direction (FMy). These are similar to those of the uniform XY chains in a transverse field (i.e., =1). When =-1, the FMx and FMy phases disappear and there appears a new FM phase. The line h=hc2=1-2 separates this new FM phase from the PM phase. The new phase is gapless with two zeros in single particle energy spectrum. This is due to the new symmetry in the system with =-1, i.e., the Hamiltonian is invariant under the transformation 2ix 2i+1y,2iy 2i+1x. The correlation function between the 2i-1 and 2i lattice points along the x (y) direction is equal to that between the 2i and 2i+1 lattice points along the y (x) direction. As a result, the long-range correlation functions along two directions are equivalent. In order to facilitate the description, we call this gapless phase the isotropic ferromagnetic (FMxx) phase. Finally, the relationship between quantum entanglement and quantum phase transitions of the system is studied. The scaling behaviour of the von Neumann entropy at each point in the FMxx phase is SL~1/3log2L+ Const, which is similar to that at the anisotropic phase transition point of the uniform XY model in a transverse field, and different from those in the FMx and FMy phases.

Acta Physica Sinica•Jan 1, 2017•1 citation

Phase diagram of the anisotropic spin- 1 2 X Z chain through the lens of bifurcation in ground-state fidelity, fidelity susceptibility, and local quantum uncertainty

Yan-Wei Dai, D. C. Liu, Xijing Liu +1 more

In this study we investigate the ground-state phase diagram of the quantum spin-1/2 anisotropic XZ chain using the infinite time-evolving block decimation algorithm. The model exhibits four distinct phases: FM_{x}, FM_{z}, AFM_{z}, and PM. In terms of the transverse magnetic field h and anisotropy parameter J, our analysis focuses on the lines h=0 and 0.5 with J≥0, where different types of quantum phase transitions (QPTs) occur. We compute the bifurcation behavior of the ground-state fidelity per lattice site. On the line h=0, continuous behavior at the bifurcation point identifies a continuous QPT between the ordered FM_{x} and FM_{z} phases at the critical point J_{c1}, with bifurcations on both sides reflecting Z_{2} symmetry breaking in both phases. On the line h=0.5, a continuous bifurcation indicates a QPT between the ordered FM_{x} and disordered PM phases at J_{c2}, with bifurcation occurring only on one side, corresponding to Z_{2} symmetry breaking in a single phase. We further compute the ground-state fidelity susceptibility per lattice site. When the ground state is obtained from random initial states, the fidelity susceptibility exhibits oscillatory behavior in the symmetry-broken phase, providing a clear signature of ground-state degeneracy. These fidelity-based measures serve as effective tools for identifying critical points and characterizing the nature of symmetry breaking. To complement these results, we evaluate the two-spin Wigner-Yanase skew information and local quantum uncertainty (LQU) as indicators of quantum correlations. While both detect QPTs, we find that not all nonanalytic features in the LQU correspond to genuine phase transitions. A critical behavior analysis is carried out at J_{c1} and J_{c2} to extract the critical exponents η^{I}, η^{C}, η^{x}, and η^{z}, associated with mutual information I(r), classical correlation C(r), and spin-spin correlations 〈σ_{i}^{x}σ_{j}^{x}〉, 〈σ_{i}^{z}σ_{j}^{z}〉. Our results demonstrate that η^{I}=η^{C} and η^{x/z}=η^{α}/2 for α=I,C, consistent with known universal scaling relations. We also extract the order parameter exponent β, correlation length exponent ν, dynamic exponent z, and the central charge c. A transition at J_{c1} between the FM_{x} and FM_{z} phases is identified as a Gaussian transition with central charge c=1, involving the breaking of two Z_{2} symmetries and lying beyond the conventional Landau-Ginzburg-Wilson symmetry-breaking paradigm. The transition at J_{c2} between the FM_{x} and PM phases belongs to the quantum Ising universality class, characterized by c=1/2 and breaking of a single Z_{2} symmetry.

Physical review. E•Feb 17, 2026•1 citation

Colorful points in the XY regime of XXZ quantum magnets

Santanu Pal, P. A. Sharma, Hitesh J. Changlani +1 more

In the XY regime of the XXZ Heisenberg model phase diagram, we demonstrate that the origin of magnetically ordered phases is influenced by the presence of solvable points with exact quantum coloring ground states featuring a quantum-classical correspondence. Using exact diagonalization and density matrix renormalization group calculations, for both the square and the triangular lattice magnets, we show that the ordered physics of the solvable points in the extreme XY regime, at $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}1$ and $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}\frac{1}{2}$, respectively, with ${J}_{\ensuremath{\perp}}>0$, adiabatically extends to the more isotropic regime $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}\ensuremath{\sim}1$. We highlight the projective structure of the coloring ground states to compute the correlators in fixed magnetization sectors which enables an understanding of the features in the static spin structure factors and correlation ratios. These findings are contrasted with an anisotropic generalization of the celebrated one-dimensional Majumdar-Ghosh model, which is also found to be (ground-state) solvable. For this model, both exact dimer and three-coloring ground states exist at $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}\frac{1}{2}$ but only the two dimer ground states survive for any $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}>\ensuremath{-}\frac{1}{2}$.

arXiv:2012.05930•Physical review. B./Physical review. B•Apr 9, 2021•13 citations

Exotic criticality in the dimerized spin-1 $XXZ$ chain with single-ion anisotropy

Satoshi Ejima, Tomoki Yamaguchi, Fabian H. L. Eßler +3 more

We consider the dimerized spin-1 XXZ chain with single-ion anisotropy D. In absence of an explicit dimerization there are three phases: a large-$D$, an antiferromagnetically ordered and a Haldane phase. This phase structure persists up to a critical dimerization, above which the Haldane phase disappears. We show that for weak dimerization the phases are separated by Gaussian and Ising quantum phase transitions. One of the Ising transitions terminates in a critical point in the universality class of the dilute Ising model. We comment on the relevance of our results to experiments on quasi-one-dimensional anisotropic spin-1 quantum magnets.

arXiv:1810.02195•SciPost Physics•Dec 7, 2018•21 citations
We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by direct generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic regime. This model also supports three different ferromagnetic (FM) phases, denoted as xxx-FM, yyy-FM and zzz-FM. The boundaries between the dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry or Z2\mathbb{Z}_2Z2​ symmetry lead to dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge c=1c=1c=1. We characterize the properties of these boundaries using entanglement entropy, excitation gap and long-range spin-spin correlations. These results are relevant to a large number of quasi-one-dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in true materials with magnetic field along different directions, and show that the anisotropy may resolve the long-standing disagreements between theory and experiments.
We report the exact dimer phase, in which the ground states are described by product of singlet dimer, in the extended XYZ model by generalizing the isotropic Majumdar-Ghosh model to the fully anisotropic region. We demonstrate that this phase can be realized even in models when antiferromagnetic interaction along one of the three directions. This model also supports three different ferromagnetic (FM) phases, denoted as x-FM, y-FM and z-FM, polarized along the three directions. The boundaries between the exact dimer phase and FM phases are infinite-fold degenerate. The breaking of this infinite-fold degeneracy by either translational symmetry breaking or [Formula: see text] symmetry breaking leads to exact dimer phase and FM phases, respectively. Moreover, the boundaries between the three FM phases are critical with central charge [Formula: see text] for free fermions. We characterize the properties of these boundaries using entanglement entropy, excitation gap, and long-range spin-spin correlation functions. These results are relevant to a large number of one dimensional magnets, in which anisotropy is necessary to isolate a single chain out from the bulk material. We discuss the possible experimental signatures in realistic materials with magnetic field along different directions and show that the anisotropy may resolve the disagreement between theory and experiments based on isotropic spin-spin interactions.
Interacting spin chains and ladders are known to support a plethora of quantum phases with complex ground-state phase diagrams. In this work, we study a large family of such models and determine precise, explicit conditions under which an exact dimer state is guaranteed to be the ground state. These general conditions are validated for various generalizations of the Majumdar-Ghosh model using exact diagonalization. Our results provide exact reference points in the phase diagrams of a wide class of spin chains and ladders, including those with anisotropic and arbitrary-range interactions.
The quantum phase transitions of one-dimensional period-two anisotropic XY models in a transverse field with the Hamiltonian where the anisotropy parameters i take and alternately, are studied. The Hamiltonian can be reduced to the diagonal form by Jordan-Wigner and Bogoliubov transformations. The long-range correlations Cx and Cy are calculated numerically. The phase with Cx Cy0 (or Cy Cx0) is referred to as the ferromagnetic (FM) phase along the x (or y) direction, while the phase with Cx=Cy=0 is the paramagnetic (PM) phase. It is found that the phase diagrams with the ratio -1 and =-1 are different obviously. For the case with -1, the line h=hc1=1-[(1-)/2]2 separates an FM phase from a PM phase, while the line =0 is the boundary between a ferromagnetic phase along the x direction (FMx) and a ferromagnetic phase along the y direction (FMy). These are similar to those of the uniform XY chains in a transverse field (i.e., =1). When =-1, the FMx and FMy phases disappear and there appears a new FM phase. The line h=hc2=1-2 separates this new FM phase from the PM phase. The new phase is gapless with two zeros in single particle energy spectrum. This is due to the new symmetry in the system with =-1, i.e., the Hamiltonian is invariant under the transformation 2ix 2i+1y,2iy 2i+1x. The correlation function between the 2i-1 and 2i lattice points along the x (y) direction is equal to that between the 2i and 2i+1 lattice points along the y (x) direction. As a result, the long-range correlation functions along two directions are equivalent. In order to facilitate the description, we call this gapless phase the isotropic ferromagnetic (FMxx) phase. Finally, the relationship between quantum entanglement and quantum phase transitions of the system is studied. The scaling behaviour of the von Neumann entropy at each point in the FMxx phase is SL~1/3log2L+ Const, which is similar to that at the anisotropic phase transition point of the uniform XY model in a transverse field, and different from those in the FMx and FMy phases.
In this study we investigate the ground-state phase diagram of the quantum spin-1/2 anisotropic XZ chain using the infinite time-evolving block decimation algorithm. The model exhibits four distinct phases: FM_{x}, FM_{z}, AFM_{z}, and PM. In terms of the transverse magnetic field h and anisotropy parameter J, our analysis focuses on the lines h=0 and 0.5 with J≥0, where different types of quantum phase transitions (QPTs) occur. We compute the bifurcation behavior of the ground-state fidelity per lattice site. On the line h=0, continuous behavior at the bifurcation point identifies a continuous QPT between the ordered FM_{x} and FM_{z} phases at the critical point J_{c1}, with bifurcations on both sides reflecting Z_{2} symmetry breaking in both phases. On the line h=0.5, a continuous bifurcation indicates a QPT between the ordered FM_{x} and disordered PM phases at J_{c2}, with bifurcation occurring only on one side, corresponding to Z_{2} symmetry breaking in a single phase. We further compute the ground-state fidelity susceptibility per lattice site. When the ground state is obtained from random initial states, the fidelity susceptibility exhibits oscillatory behavior in the symmetry-broken phase, providing a clear signature of ground-state degeneracy. These fidelity-based measures serve as effective tools for identifying critical points and characterizing the nature of symmetry breaking. To complement these results, we evaluate the two-spin Wigner-Yanase skew information and local quantum uncertainty (LQU) as indicators of quantum correlations. While both detect QPTs, we find that not all nonanalytic features in the LQU correspond to genuine phase transitions. A critical behavior analysis is carried out at J_{c1} and J_{c2} to extract the critical exponents η^{I}, η^{C}, η^{x}, and η^{z}, associated with mutual information I(r), classical correlation C(r), and spin-spin correlations 〈σ_{i}^{x}σ_{j}^{x}〉, 〈σ_{i}^{z}σ_{j}^{z}〉. Our results demonstrate that η^{I}=η^{C} and η^{x/z}=η^{α}/2 for α=I,C, consistent with known universal scaling relations. We also extract the order parameter exponent β, correlation length exponent ν, dynamic exponent z, and the central charge c. A transition at J_{c1} between the FM_{x} and FM_{z} phases is identified as a Gaussian transition with central charge c=1, involving the breaking of two Z_{2} symmetries and lying beyond the conventional Landau-Ginzburg-Wilson symmetry-breaking paradigm. The transition at J_{c2} between the FM_{x} and PM phases belongs to the quantum Ising universality class, characterized by c=1/2 and breaking of a single Z_{2} symmetry.
In the XY regime of the XXZ Heisenberg model phase diagram, we demonstrate that the origin of magnetically ordered phases is influenced by the presence of solvable points with exact quantum coloring ground states featuring a quantum-classical correspondence. Using exact diagonalization and density matrix renormalization group calculations, for both the square and the triangular lattice magnets, we show that the ordered physics of the solvable points in the extreme XY regime, at $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}1$ and $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}\frac{1}{2}$, respectively, with ${J}_{\ensuremath{\perp}}>0$, adiabatically extends to the more isotropic regime $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}\ensuremath{\sim}1$. We highlight the projective structure of the coloring ground states to compute the correlators in fixed magnetization sectors which enables an understanding of the features in the static spin structure factors and correlation ratios. These findings are contrasted with an anisotropic generalization of the celebrated one-dimensional Majumdar-Ghosh model, which is also found to be (ground-state) solvable. For this model, both exact dimer and three-coloring ground states exist at $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}=\ensuremath{-}\frac{1}{2}$ but only the two dimer ground states survive for any $\frac{{J}_{z}}{{J}_{\ensuremath{\perp}}}>\ensuremath{-}\frac{1}{2}$.
Exotic criticality in the dimerized spin-1 XXZXXZXXZ chain with single-ion anisotropy
We consider the dimerized spin-1 XXZ chain with single-ion anisotropy D. In absence of an explicit dimerization there are three phases: a large-DDD, an antiferromagnetically ordered and a Haldane phase. This phase structure persists up to a critical dimerization, above which the Haldane phase disappears. We show that for weak dimerization the phases are separated by Gaussian and Ising quantum phase transitions. One of the Ising transitions terminates in a critical point in the universality class of the dilute Ising model. We comment on the relevance of our results to experiments on quasi-one-dimensional anisotropic spin-1 quantum magnets.